JEE Challenger
More from Binomial Theorem
MathMedMain 2026 • 2nd Apr - Shift 2

Simplify Sum of Binomial Coefficients to Find Value of m

If for 3≤r≤303 \le r \le 30, (30C30−r)+3(30C31−r)+3(30C32−r)+(30C33−r)=mCr\left(^{30}C_{30-r}\right) + 3\left(^{30}C_{31-r}\right) + 3\left(^{30}C_{32-r}\right) + \left(^{30}C_{33-r}\right) = {^m}C_r, then mm equals :

Options

A

31

B

32

C

33

Correct
D

34

Topics & Concepts

Binomial TheoremBinomial Theorem for Positive Integral Index

Step-by-Step Solution

To find the value of mm, we simplify the expression given on the left-hand side (LHS).

Using the property of binomial coefficients, nCk=nCn−k^nC_k = ^nC_{n-k}, we can rewrite each term as follows:

  1. 30C30−r=30C30−(30−r)=30Cr^{30}C_{30-r} = ^{30}C_{30-(30-r)} = ^{30}C_r
  2. 30C31−r=30C30−(31−r)=30Cr−1^{30}C_{31-r} = ^{30}C_{30-(31-r)} = ^{30}C_{r-1}
  3. 30C32−r=30C30−(32−r)=30Cr−2^{30}C_{32-r} = ^{30}C_{30-(32-r)} = ^{30}C_{r-2}
  4. 30C33−r=30C30−(33−r)=30Cr−3^{30}C_{33-r} = ^{30}C_{30-(33-r)} = ^{30}C_{r-3}

Substituting these back into the LHS, we get: LHS=30Cr+3(30Cr−1)+3(30Cr−2)+30Cr−3\text{LHS} = ^{30}C_r + 3\left(^{30}C_{r-1}\right) + 3\left(^{30}C_{r-2}\right) + ^{30}C_{r-3}

We apply Pascal's identity, which states that nCk+nCk−1=n+1Ck^nC_k + ^nC_{k-1} = ^{n+1}C_k, by grouping the terms in steps:

Step 1: Split the coefficients: LHS=(30Cr+30Cr−1)+2(30Cr−1+30Cr−2)+(30Cr−2+30Cr−3)\text{LHS} = \left(^{30}C_r + ^{30}C_{r-1}\right) + 2\left(^{30}C_{r-1} + ^{30}C_{r-2}\right) + \left(^{30}C_{r-2} + ^{30}C_{r-3}\right)

Applying Pascal's identity to each pair: LHS=31Cr+2(31Cr−1)+31Cr−2\text{LHS} = ^{31}C_r + 2\left(^{31}C_{r-1}\right) + ^{31}C_{r-2}

Step 2: Split and group again: LHS=(31Cr+31Cr−1)+(31Cr−1+31Cr−2)\text{LHS} = \left(^{31}C_r + ^{31}C_{r-1}\right) + \left(^{31}C_{r-1} + ^{31}C_{r-2}\right)

Applying Pascal's identity again: LHS=32Cr+32Cr−1\text{LHS} = ^{32}C_r + ^{32}C_{r-1}

Step 3: Apply Pascal's identity one last time: LHS=33Cr\text{LHS} = ^{33}C_r

Given that LHS=mCr\text{LHS} = {}^mC_r, we have: 33Cr=mCr^{33}C_r = {}^mC_r

Therefore, m=33m = 33.

Simplify Sum of Binomial Coefficients to Find Value of m | Mathematics PYQ Solution - JEE Challenger